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Advances in Acoustics and Vibration
Volume 2011 (2011), Article ID 159437, 5 pages
http://dx.doi.org/10.1155/2011/159437
Research Article

Propagation of Waves at an Interface between a Liquid Half-Space and an Orthotropic Micropolar Solid Half-Space

1Department of Mathematics, Post Graduate Government College, Sector 11, Chandigarh 160 011, India
2Department of Mathematics, S. R. M. Institute of Engineering and Technology, Haryana, Khora-Bhura 134203, India

Received 16 May 2011; Accepted 23 July 2011

Academic Editor: K. M. Liew

Copyright © 2011 Baljeet Singh and Ritu Sindhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An inviscid liquid half-space is considered in welded contact with a orthotropic micropolar solid half-space. Appropriate plane harmonic solutions of equations governing a liquid half-space and an orthotropic solid half-space are obtained. These solutions satisfy the required boundary conditions at the interface to obtain a system of four nonhomogeneous equations in amplitude ratios for incident quasi-longitudinal displacement wave. The amplitude ratios of various reflected and refracted waves are computed numerically for a particular example of the present model. The effect of anisotropy upon these amplitude ratios is shown graphically for a particular range of the angle of incidence.

1. Introduction

Material response to external stimuli depends heavily on the motions of its inner structures. Classical elasticity does not include this effect, where only translation degrees of freedom of material point of body is considered. Eringen [1] developed the linear micropolar theory of elasticity, which included the intrinsic rotations of the microstructure. It provides a model which can support body and surface couples and display high frequency optical branch of the wave spectrum. For engineering applications, it can model composites with rigid chopped fibres, elastic solid with rigid granular inclusions, and other industrial materials such as liquid crystals.

Parfitt and Eringen [2] investigated the plane wave propagation in an infinite isotropic homogeneous micropolar elastic solid half-space and showed the existence of four basic waves (a longitudinal displacement wave, a longitudinal microrotational wave, and two sets of two coupled waves) propagating with different velocities in an isotropic micropolar elastic solid. Smith [3] and Ariman [4] also studied the wave propagation in micropolar elastic solids.

The assumptions of isotropy in the solid medium may not capture some significant features of the continuum responses of soils, geological materials, and composites. Iesan [57] studied some static problems in orthotropic micropolar elastic solids. Kumar and Choudhary [8, 9] studied the mechanical sources and dynamic behaviour of orthotropic micropolar elastic medium. Kumar and Chaudhary [10] explained the plane strain problem in a homogeneous micropolar orthotropic elastic solids. Kumar and Ailawalia [11] studied the response of a micropolar cubic crystal due to various sources. Kumar and Gupta [12] studied the propagation of waves in transversely isotropic micropolar generalized thermoelastic half-space. Singh [13] investigated the two-dimensional plane wave propagation in an orthotropic micropolar elastic soli and showed the existence of three types of coupled plane waves in xy-plane, whose velocities depend upon the angle of propagation and material parameters. He also studied the reflection of these plane waves from a stress-free free surface and obtained the reflection coefficients for various reflected waves.

The study of wave motions at liquid-solid interface has been a topic of research for the last many years. Recently, Singh [14] studied the reflection and transmission of plane harmonic waves at an interface between liquid and micropolar viscoelastic solid with stretch, where the effects of micropolarity, viscosity, and stretch are observed on reflection and transmission coefficients. In the present paper, we have considered a problem on reflection and transmission at an interface between an inviscid liquid half-space and an orthotropic solid half-space. The reflection and transmission coefficients are computed numerically for a particular model to observe the effect of orthotropy.

2. Formulation of the Problem and Solution

We consider a homogeneous and orthotropic medium of an infinite extent with Cartesian coordinate system (𝑥1,𝑥2,𝑥3). We restrict our study to the plane strain parallel to 𝑥1𝑥2-plane, with the displacement vector 𝐮=(𝑢1,𝑢2,0) and microrotation vector 𝜙=(0,0,𝜙3). Following Eringen [15] and Iesan [5], the field equations for homogeneous and orthotropic micropolar solid in 𝑥1𝑥2-plane in absence of body forces and couples are given by 𝐴11𝑢1,11+𝐴12+𝐴78𝑢2,12+𝐴88𝑢1,22𝐾1𝜙3,2=𝜌̈𝑢1,(1)𝐴12+𝐴78𝑢1,12+𝐴77𝑢2,11+𝐴22𝑢2,22𝐾2𝜙3,1=𝜌̈𝑢2,(2)𝐵66𝜙3,11+𝐵44𝜙3,22𝜒𝜙3+𝐾1𝑢1,2+𝐾2𝑢2,1̈𝜙=𝜌𝑗3,(3) where 𝐴11,𝐴12,𝐴22,𝐴77,𝐴78,𝐴88,𝐵44,𝐵66 are characteristic constants of the material, 𝜌 is the density of the medium, and 𝑗 is the microinertia. Here, 𝜒=𝐾2𝐾1,𝐾1=𝐴78𝐴88,𝐾2=𝐴77𝐴78,̈𝑢𝑖=(𝜕2𝑢𝑖/𝜕𝑡2),𝑢𝑖,𝑗𝑘=(𝜕2𝑢𝑖/𝜕𝑥𝑗𝜕𝑥𝑘), and ̈𝜙𝑖=(𝜕2𝜙𝑖/𝜕𝑡2).

Solutions of (1) to (3) are now sought in the form of the harmonic travelling wave:𝑢𝑖,𝜙3=𝐴𝑑𝑖𝑒,𝑘𝐵𝜄𝑘(𝑥1𝑝1+𝑥2𝑝2𝑣𝑡),(𝑖=1,2),(4) where 𝑘 is the wave number, 𝑣 is the phase velocity, 𝑑𝑖 are components of unit displacement vector, 𝑝𝑗(𝑗=1,2) are components of propagation vector, and 𝐴,𝐵 are arbitrary constants.

Using (4), (1) to (3) reduce to 𝐷1𝜌𝑣2𝐴𝑑1+𝐴12+𝐴78𝑝1𝑝2𝐴𝑑2+𝜄𝑝2𝐾1𝐴𝐵=0,12+𝐴78𝑝1𝑝2𝐴𝑑1+𝐷2𝜌𝑣2𝐴𝑑2+𝜄𝑝1𝐾2𝑝𝐵=0,2𝐾1𝐴𝑑1+𝑝1𝐾2𝐴𝑑2+𝜄𝜒+𝑗𝑘2𝐷3𝜌𝑣2𝐵=0,(5) where 𝐷1=𝐴11𝑝12+𝐴88𝑝22,𝐷2=𝐴77𝑝12+𝐴22𝑝22, and 𝐷3=(𝐵66𝑝12+𝐵44𝑝22)/𝑗. The nontrivial solution of (5) gives the following cubic equation in 𝜁:𝜁3+𝐿𝜁2+𝑀𝜁+𝑁=0,(6) where 𝜁=𝜌𝑣2,𝐿=[𝐷1+𝐷2+𝐷3+(𝜒/𝑗𝑘2)],  𝑀=𝐷1𝐷2+𝐷2𝐷3+𝐷3𝐷1+(𝐷1+𝐷2)(𝜒/𝑗𝑘2)(1/𝑗𝑘2)(𝑝12𝐾22+𝑝22𝐾12)(𝐴12+𝐴78)2𝑝12𝑝22,  𝑁=[𝐷1𝐷2𝐷3+(𝜒/𝑗𝑘2){𝐷1𝐷2(𝐴12+𝐴78)2𝑝12𝑝22}(1/𝑗𝑘2){𝐷1𝐾22𝑝12+𝐷2𝐾12𝑝222𝐾1𝐾2(𝐴12+𝐴78)𝑝12𝑝22}𝐷3(𝐴12+𝐴78)2𝑝12𝑝22] The three roots 𝜁1,𝜁2, and 𝜁3 of the cubic equation (6) correspond to three types of quasi-coupled waves, whose velocities depend upon the angle of propagation. Let us name these three waves as quasi-longitudinal displacement (QLD) wave, quasi coupled transverse microrotational (QCTM) wave and quasi coupled transverse displacement (QCTD) wave. The corresponding velocities of these waves are identified from computer program of the solution of (6). It is found that the velocities 𝑣1,𝑣2, and 𝑣3, correspond to QLD, QCTM, and QCTD waves, respectively [13].

If we put 𝐴11=𝐴22=𝜆+2𝜇+𝜅,𝐴77=𝐴88=𝜇+𝜅,𝐴12=𝜆,𝐴78=𝜇,𝐾1=𝐾2=𝜒/2=𝜅, in (6), then the velocity 𝑣1 corresponds to longitudinal displacement wave and the velocities 𝑣2,𝑣3 correspond to two coupled transverse waves as obtained by Parfitt and Eringen [2] in the theory of isotropic and linear micropolar elasticity. These velocities do not depend upon the angle of propagation.

If we put 𝐴77=𝐴88=𝐴78=𝐵44=𝐵66=𝐾1=𝐾2=𝜒=0,𝐴11=𝐴22=𝐴12=𝜆 in (6), the cubic equation (6) reduces to the linear equation 𝜌𝑣2=𝜆, which gives the velocity of longitudinal wave in the liquid medium. Here, the parameters with primes correspond to the liquid medium.

3. Reflection and Transmission

We consider a homogeneous and orthotropic micropolar medium and a liquid medium of an infinite extent with cartesian coordinate system (𝑥1,𝑥2,𝑥3) having origin at the interface. The positive 𝑥3-axis is pointing into the orthotropic micropolar solid half-space. The 𝑥1-axis is along the interface between the liquid half-space and orthotropic micropolar solid half-space. In this section, we shall derive the relations between the reflection and transmission coefficients, when (QLD, QCTM, or QCTD) wave becomes incident at the interface 𝑥2=0. For incident (QLD, QCTM, or QCTD) wave, there will be reflected QLD, QCTM, and QCTD waves in solid half-space and transmitted longitudinal wave in the liquid half-space as shown in Figure 1. Accordingly, if the wave normal to the incident wave through solid medium makes an angle 𝜃0 with the positive direction of 𝑥2-axis, then wave normals to the reflected QLD, QCTM, and QCTD and refracted longitudinal waves make 𝜃1,𝜃2, and 𝜃3 and and 𝜃 with 𝑥2-axis. The required boundary conditions at the interface, 𝑥2=0, are the continuity of normal force stresses, vanishing of tangential force stress, vanishing of tangential couple stress, and continuity of normal components of displacement vector, that is, 𝑡22(𝛼)=𝑡22(𝛼),𝑡21(𝛼)=0,𝑚23(𝛼)𝑢=0,2(𝛼)=𝑢2(7) where 𝑡22(𝛼)=𝐴12𝑢1,1+𝐴22𝑢2,2,𝑡21(𝛼)=𝐴78𝑢2,1+𝐴88𝑢1,2+𝐴88𝐴78𝜙3,𝑚23(𝛼)=𝐵44𝜙3,2,𝑡22=𝜆𝑢1,1+𝑢2,2.(8) The following components of displacement and microrotation vectors are appropriate to satisfy the boundary conditions (7) 𝑢1(𝛼)=𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝜂𝛼,𝑢2(𝛼)=𝐺(𝛼)𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝜂𝛼,𝜙3(𝛼)=𝑖𝑘𝛼𝐻(𝛼)𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝜂𝛼,𝑢2=𝐺𝐴𝑑1𝑒𝑖𝜂.(9) Here, 𝜂𝛼=𝑘𝛼(𝑥𝑝1(𝛼)+𝑦𝑝2(𝛼)𝑣𝛼𝑡),𝛼=0 corresponds to incident wave; 𝛼=1,2,3 correspond to reflected waves, and 𝐺(𝛼)=𝐾1𝐴12+𝐴78𝑝1(𝛼)𝑝22(𝛼)𝐾2𝑝1(𝛼)𝐷1(𝛼)𝜌𝑣𝛼2𝐾2𝐴12+𝐴78𝑝2(𝛼)𝑝12(𝛼)𝐾1𝑝2(𝛼)𝐷2(𝛼)𝜌𝑣𝛼2,𝐻(𝛼)=𝐾1𝑝2(𝛼)+𝐾2𝐺(𝛼)𝑝1(𝛼)𝜒+𝑗𝑘2𝐷3(𝛼)𝜌𝑣𝛼2,(10) where 𝐷1(𝛼)=𝐴11𝑝12(𝛼)+𝐴88𝑝22(𝛼),𝐷2(𝛼)=𝐴77𝑝12(𝛼)+𝐴22𝑝22(𝛼),𝐷3(𝛼)=𝐵66𝑝12(𝛼)+𝐵44𝑝22(𝛼)𝑗,(11) and 𝐺 and 𝜂 correspond to the liquid medium.

159437.fig.001
Figure 1: Geometry of the problem.

With the help of above displacement and microrotation components, the boundary conditions (7) result into the following three relations𝑘𝛼𝐴12𝑝1(𝛼)+𝐴22𝐺(𝛼)𝑝2(𝛼)𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝑘𝛼(𝑥𝑝1(𝛼)𝑣𝛼𝑡)𝑘𝜆𝑝1+𝐺𝑝2𝐴𝑑1𝑒𝑖𝑘(𝑥1𝑝1𝑣𝑡)𝐴=0,78𝑝1(𝛼)𝐺(𝛼)+𝐴88𝑝2(𝛼)+𝐴88𝐴78𝐻(𝛼)×𝑘𝛼𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝑘𝛼(𝑥𝑝1(𝛼)𝑣𝛼𝑡)𝑘=0,𝛼2𝐻(𝛼)𝑝2(𝛼)𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝑘𝛼(𝑥𝑝1(𝛼)𝑣𝛼𝑡)𝐺=0,(𝛼)𝐴(𝛼)𝑑1(𝛼)𝑒𝑖𝑘𝛼(𝑥1𝑝1(𝛼)𝑣𝛼𝑡)𝐺𝐴𝑑1𝑒𝑖𝑘(𝑥1𝑝1𝑣𝑡)=0.(12) In view of the fact that (12) must hold for all 𝑥 and 𝑡, it is deduced that 𝑘0𝑝1(0)=𝑘1𝑝1(1)=𝑘2𝑝1(2)=𝑘3𝑝1(3)=𝑘𝑝1𝑘𝑘,(13)0𝑣0=𝑘1𝑣1=𝑘2𝑣2=𝑘3𝑣3=𝑘𝑣𝜔,(14) From (12), the analytical expressions 𝑍1=𝐴(1)/𝐴(0),𝑍2=𝐴(2)/𝐴(0),𝑍3=𝐴(3)/𝐴(0), and 𝑍4=𝐴/𝐴(0) for reflection coefficients of QLD, QCTM, and QCTD waves and transmission coefficient of longitudinal wave may be obtained for further numerical computations.

4. Numerical Example

The following relevant physical constants are chosen arbitrarily for a composite as an orthotropic micropolar material due to unavailability of relevant experimental data for such material in literature [8].

𝐴11=11.65×1010Nm2,𝐴22=11.71×1010Nm2,𝐴12=7.69×1010Nm2,𝐴77=1.9959×1010Nm2,𝐴78=1.98×1010Nm2,𝐴88=2.0159×1010Nm2,𝐵44=0.036×1010N,𝐵66=0.037×1010N,𝜌=2190kgm3,𝑗=0.000196m2,𝑗𝑘2=1. The following physical constants for aluminium-epoxy composite as an isotropic micropolar elastic solid are considered [16]: 𝜌=2190kgm3,𝜆=7.59×1010Nm2,𝜇=1.89×1010Nm2,𝜅=0.0149×1010Nm2,𝛾=0.0268×1010N,𝑗=0.000196m2,𝑗𝑘2=1.

The following physical constants for inviscid liquid are considered 𝜌=1000kgm3,𝑣=1.435×104ms1,

For the incidence of QLD wave, the complex absolute values of the reflection and transmission coefficients are computed numerically with the help of above physical constants of orthotropic and isotropic micropolar materials with the following components of the propagation and unit displacement vectors:𝑝1(0)=sin𝜃0,𝑝2(0)=cos𝜃0,𝑝1(1)=sin𝜃1,𝑝2(1)=cos𝜃1,𝑑1(0)=cos𝜃0,𝑑2(0)=sin𝜃0,𝑑1(1)=cos𝜃1,𝑑2(1)=sin𝜃1,𝑝1(2)=sin𝜃2,𝑝2(2)=cos𝜃2,𝑝1(3)=sin𝜃3,𝑝2(3)=cos𝜃3,𝑑1(2)=cos𝜃2,𝑑2(2)=sin𝜃2,𝑑1(3)=cos𝜃3,𝑑2(3)=sin𝜃3,𝑝1=sin𝜃,𝑝2=cos𝜃,𝑑1=cos𝜃,𝑑2=sin𝜃.(15) The reflection coefficient 𝑍1 of reflected QLD wave has value 0.4182 near normal incidence. It decreases to its minimum value 0.2554 at 𝜃0=53; thereafter, it increases to its maximum value 0.9447 near grazing incidence. The reflection coefficient 𝑍2 of reflected QCTM wave has value 0.0606 near normal incidence. Initially, it increases to its maximum value 0.0705 at 𝜃0=3, but then it sharply decreases to 0.0011 at 𝜃0=45. Thereafter, it increases to value 0.0265 at 𝜃0=65, and then decreases to its minimum value 0.0001 at 𝜃0=89. The reflection coefficient 𝑍3 of QCTD wave has its maximum value 1.1057 at 𝜃0=1. It decreases slowly to its minimum value 0.0009 near grazing incidence. The transmission coefficient 𝑍4 of longitudinal wave has its maximum value 0.5837 at 𝜃0=1. It decreases slowly to its minimum value 0.0004 near grazing incidence. These variations of reflection and transmission coefficients are shown in Figure 2 by solid lines. The dotted lines in Figure 2 correspond to the variations for isotropic micropolar case. On comparing the solid and dotted lines, it is observed that the reflection coefficients of reflected QCTM and QCTD waves are significantly affected due to the presence of orthotropy in micropolar medium. The coefficients of reflected QLD and refracted longitudinal waves are also affected slightly due to the anisotropy in the solid medium.

fig2
Figure 2: Variations of amplitude ratios of reflected QLD, QCTM, QCTD and refracted longitudinal waves against the angle of incidence.

5. Concluding Remarks

The relevant boundary conditions at an interface between the liquid half-space and orthotropic micropolar solid half-space interface are satisfied by appropriate solutions in the both half-spaces to obtain the relations between reflection and transmission coefficients of various reflected and refracted waves for the incidence of QLD, QCTM, or QCTD wave. The numerical computations are carried out for the incidence of QLD wave only. The dependence of reflection and transmission coefficients on the angle of incidence is shown graphically to observe the effect of orthotropy in solid medium. The present numerical study might provide more relevant information about the wave propagation in orthotropic material if we had relevant experimental physical data for an orthotropic micropolar material. The present theoretical and numerical analysis may be helpful to experimental seismologists working in the fields of wave propagation in solids.

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